Pressure

Pressure ($p$) is the force per unit area applied on a surface, in a direction perpendicular to that surface, i.e. the scalar part of the stress tensor under equilibrium/hydrostatic conditions.

Thermodynamics

In thermodynamics the pressure is given by

$p=-\left.{\frac {\partial A}{\partial V}}\right\vert _{T,N}=k_{B}T\left.{\frac {\partial \ln Q}{\partial V}}\right\vert _{T,N}$

where $A$ is the Helmholtz energy function, $V$ is the volume, $k_{B}$ is the Boltzmann constant, $T$ is the temperature and $Q(N,V,T)$ is the canonical ensemble partition function.

Units

The SI units for pressure are Pascals (Pa), 1 Pa being 1 N/m², or 1 J/m³. Other frequently encountered units are bars and millibars (mbar); 1 mbar = 100 Pa = 1 hPa, 1 hectopascal. 1 bar is 10⁵ Pa by definition. This is very close to the standard atmosphere (atm), approximately equal to typical air pressure at earth mean sea level: atm, standard atmosphere = 101325 Pa = 101.325 kPa = 1013.25 hPa = 1.01325 bar

Stress

The stress is given by

${\mathbf {F} }=\sigma _{ij}{\mathbf {A} }$

where ${\mathbf {F} }$ is the force, ${\mathbf {A} }$ is the area, and $\sigma _{ij}$ is the stress tensor, given by

$\sigma _{ij}\equiv \left[{\begin{matrix}\sigma _{x}&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\sigma _{y}&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\sigma _{z}\\\end{matrix}}\right]$

where where $\ \sigma _{x}$, $\ \sigma _{y}$, and $\ \sigma _{z}$ are normal stresses, and $\ \tau _{xy}$, $\ \tau _{xz}$, $\ \tau _{yx}$, $\ \tau _{yz}$, $\ \tau _{zx}$, and $\ \tau _{zy}$ are shear stresess.

Virial pressure

The virial pressure is commonly used to obtain the pressure from a general simulation. It is particularly well suited to molecular dynamics, since forces are evaluated and readily available. For pair interactions, one has (Eq. 2 in ):

$p={\frac {k_{B}TN}{V}}+{\frac {1}{Vd}}{\overline {\sum _{i<j}{\mathbf {f} }_{ij}{\mathbf {r} }_{ij}}},$

where $p$ is the pressure, $T$ is the temperature, $V$ is the volume and $k_{B}$ is the Boltzmann constant. In this equation one can recognize an ideal gas contribution, and a second term due to the virial. The overline is an average, which would be a time average in molecular dynamics, or an ensemble average in Monte Carlo; $d$ is the dimension of the system (3 in the "real" world). ${\mathbf {f} }_{ij}$ is the force on particle $i$ exerted by particle $j$, and ${\mathbf {r} }_{ij}$ is the vector going from $i$ to $j$: ${\mathbf {r} }_{ij}={\mathbf {r} }_{j}-{\mathbf {r} }_{i}$.

This relationship is readily obtained by writing the partition function in "reduced coordinates", i.e. $x^{*}=x/L$, etc, then considering a "blow-up" of the system by changing the value of $L$. This would apply to a simple cubic system, but the same ideas can also be applied to obtain expressions for the stress tensor and the surface tension, and are also used in constant-pressure Monte Carlo.

If the interaction is central, the force is given by

${\mathbf {f} }_{ij}=-{\frac {{\mathbf {r} }_{ij}}{r_{ij}}}f(r_{ij}),$

where $f(r)$ the force corresponding to the intermolecular potential $\Phi (r)$:

$-\partial \Phi (r)/\partial r.$

For example, for the Lennard-Jones potential, $f(r)=24\epsilon (2(\sigma /r)^{12}-(\sigma /r)^{6})/r$. Hence, the expression reduces to

$p={\frac {k_{B}TN}{V}}+{\frac {1}{Vd}}{\overline {\sum _{i<j}f(r_{ij})r_{ij}}}.$

Notice that most realistic potentials are attractive at long ranges; hence the first correction to the ideal pressure will be a negative contribution: the second virial coefficient. On the other hand, contributions from purely repulsive potentials, such as hard spheres, are always positive.

Pressure equation

For particles acting through two-body central forces alone one may use the thermodynamic relation

$p=-\left.{\frac {\partial A}{\partial V}}\right\vert _{T}$

Using this relation, along with the Helmholtz energy function and the canonical partition function, one arrives at the so-called pressure equation (also known as the virial equation):

$p^{*}={\frac {\beta p}{\rho }}={\frac {pV}{Nk_{B}T}}=1-\beta {\frac {2}{3}}\pi \rho \int _{0}^{\infty }\left({\frac {{\rm {d}}\Phi (r)}{{\rm {d}}r}}~r\right)~{\rm {g}}(r)r^{2}~{\rm {d}}r$

where $\beta :=1/k_{B}T$, $\Phi (r)$ is a central potential and ${\rm {g}}(r)$ is the pair distribution function.

See also

• Barostats
• Test volume method

References

Related reading

• Aidan P. Thompson, Steven J. Plimpton, and William Mattson "General formulation of pressure and stress tensor for arbitrary many-body interaction potentials under periodic boundary conditions", Journal of Chemical Physics 131 154107 (2009)
• G. C. Rossi and M. Testa "The stress tensor in thermodynamics and statistical mechanics", Journal of Chemical Physics 132 074902 (2010)
• Nikhil Chandra Admal and E. B. Tadmor "Stress and heat flux for arbitrary multibody potentials: A unified framework", Journal of Chemical Physics 134 184106 (2011)
• Takenobu Nakamura, Wataru Shinoda, and Tamio Ikeshoji "Novel numerical method for calculating the pressure tensor in spherical coordinates for molecular systems", Journal of Chemical Physics 135 094106 (2011)
• Péter T. Kiss and András Baranyai "On the pressure calculation for polarizable models in computer simulation", Journal of Chemical Physics 136 104109 (2012)
• Jerry Zhijian Yang, Xiaojie Wu, and Xiantao Li "A generalized Irving–Kirkwood formula for the calculation of stress in molecular dynamics models", Journal of Chemical Physics 137 134104 (2012)
• J. P. Wittmer, H. Xu, P. Polińska, F. Weysser, and J. Baschnagel "Communication: Pressure fluctuations in isotropic solids and fluids", Journal of Chemical Physics 138 191101 (2013)
• F. J. Martínez-Ruiz, F. J. Blas, B. Mendiboure and A. I. Moreno-Ventas Bravo "Effect of dispersive long-range corrections to the pressure tensor: The vapour-liquid interfacial properties of the Lennard-Jones system revisited", Journal of Chemical Physics 141 184701 (2014)
• Sadrul Chowdhury, Sneha Abraham, Toby Hudson and Peter Harrowell "Long range stress correlations in the inherent structures of liquids at rest", Journal of Chemical Physics 144 124508 (2016)
• Ronald E. Miller, Ellad B. Tadmor, Joshua S. Gibson, Noam Bernstein and Fabio Pavia "Molecular dynamics at constant Cauchy stress", Journal of Chemical Physics 144 184107 (2016)
• E. R. Smith, D. M. Heyes, and D. Dini "Towards the Irving-Kirkwood limit of the mechanical stress tensor", Journal of Chemical Physics 146 224109 (2017)
• Matthias Krüger, Alexandre Solon, Vincent Démery, Christian M. Rohwer, and David S. Dean "Stresses in non-equilibrium fluids: Exact formulation and coarse-grained theory", Journal of Chemical Physics 148 084503 (2018)